In atrocious conditions at St Mary's, Jose Mourinho's side ground out victory in impressive fashion with Fernando Torres, Willian and Oscar netting second-half efforts en route to a 3-0 victory.
Southampton's blank meant Cech equalled Bonetti's long-standing Chelsea record of 208 shut-outs, doing so in 457 appearances in contrast to the former England international's 729.
Cech is understandably proud to be in place to break the record in Sunday's FA Cup tie at Derby and honoured to be considered in the same bracket as Bonetti, who was nicknamed 'The Cat' for his agility.
"I think I will invite him for a drink," Cech said. "I'm stealing his record so I think this is the right moment.
"Peter is a great guy. Every time we meet at the Bridge we have a few words. He is very humble and this is why he is a proper legend.
"He is someone who enjoys the success of the club and he still loves the club. You always see him around and he's brilliant.
"It's a pleasure as a personal achievement, but I'm really glad that all those clean sheets helped us win trophies over the last nine-and-a-half years. That's what I'm most pleased about."
Cech has played an integral role in that period of unparalleled success at Stamford Bridge, with the Blues in the running for three trophies this season.
The Premier League is a key goal and the west Londoners are certainly in the hunt, sitting two points off the lead after an impressive festive period in which they only dropped points in a draw at leaders Arsenal.
Similarly impressive Christmas displays were key when Chelsea won the title the first two times under Mourinho, and Cech hopes to kick on in a similar fashion to wrap up another title.
"When you play away against a team at the top of the table then obviously the point is valuable, although we hit the bar and could have won the game," he said, referring to the 0-0 stalemate at Arsenal.
"But we made the most of this period. This was our aim. We are right there. It's up to us how we continue that now.
"We are improving game-by-game and we hope to still improve even more. This was a good performance at both ends.
"In the Christmas period it was very important not to concede goals and to score goals at the other end.
"We did really well during this very difficult period for us. We got most of the points we wanted and this was our target."
While Chelsea are upwardly mobile, Southampton have won just one of their last nine matches.
Mauricio Pochettino's side have gone from third to ninth in the standings, leaving Jack Cork hoping for a morale-boosting victory over his former club Burnley in the FA Cup third round this weekend.
"We've had a really tough month fixture-wise," Chelsea academy graduate Cork said.
"It's been tough for us - we've been lucky in some parts and unlucky in others.
"Hopefully the fixtures will be a bit more kind to us this January and we can start that with Burnley in the cup and try to get some confidence from that game.
"Maybe against Chelsea was a little bit different but recently the performances have been very good. Everton was a good performance. We were unlucky there.
"We've been playing well. I think things haven't been going our way - it's just one of those patches. Hopefully we can come out of it."